In this chapter, we studied various methods for computing one-loop corrections to the energies of classical solutions to type IIA string theory inAdS4×CP3. Previous studies which computed the one-loop correction to the folded spinning string in AdS4 found that agreement with the all-loop AdS4/CF T3 Bethe ansatz is not achieved using the standard summation prescription that was used for type IIB string theory inAdS5×S5. Rather, a new summation prescription seems to be required, which distinguishes between so-called light modes and heavy modes. We extended this investigation by analyzing the one-loop correction to the energy of a point-particle and a circular spinning string, both of which are located at the spatial origin of AdS4 and have nontrivial support in CP3. The spinning string considered in this chapter has two equal nonzero spins inCP3 and is the analogue of theSU(2) spinning string in AdS5×S5. The point-particle and spinning string are important examples to analyze because they have trivial support inAdS4 and therefore avoid theκ-symmetry issues that arise for solutions which purely have support inAdS4, such as the folded spinning string.
We used two techniques to compute the spectrum of uctuations about these solutions. One technique, called the world-sheet approach, involves expanding the GS action to quadratic order in the uctuations and computing the normal modes of the resulting action. The other technique, called the algebraic curve approach, involves computing the algebraic curve for the classical solutions and then carrying out semiclassical quantization. For the point-particle, we found that the world- sheet and algebraic curve uctuation frequencies match the spectrum of uctuations obtained in the
Penrose limit up to constant shifts. Furthermore, for the spinning string we found that the algebraic curve spectrum matches the world-sheet spectrum up to constant shifts and shifts in mode number.
In particular, the AC and WS frequencies for the spinning string both reduce to the corresponding point-particle frequencies when the winding number is set to zero and become unstable when the winding number|m| ≥2. This is familiar from theSU(2)spinning string inAdS5×S5[92, 4], which has instabilities for|m| ≥1.
Although the algebraic curve spectrum looks very similar to the world-sheet spectrum, it exhibits some important dierences. For example, we nd that the algebraic curve frequencies have at- space behavior when the mode numbern= 0. This was also found for algebraic curve frequencies in AdS5×S5. More importantly, if we compute one-loop corrections by adding up the algebraic curve frequencies using the standard summation prescription that was used in AdS5×S5, then we get a linear divergence. This is inconsistent with supersymmetry because we expect the one- loop correction to vanish for the point-particle and to be nonzero but nite for the spinning string.
We propose a new summation prescription in equation (5.38) which gives precisely these results when applied both to the algebraic curve spectrum and the world-sheet spectrum. This summation prescription has certain similarities to the one that was proposed in [89]. In particular, it also gives a one-loop correction to the folded spinning string which agrees with the all-loop Bethe ansatz. At the same time, it has some important dierences which are described in section 2.3. For example, we nd that our summation prescription generally gives more well-dened results for one-loop corrections.
In principle we can get three predictions for the one-loop correction to the spinning string (one coming from the algebraic curve and two coming from the world-sheet, because the world-sheet gives nite results using both the old summation prescription in equation (5.36) and the new summation prescription in equation (5.38)), but by expanding the one-loop corrections in the large-J limit (where J = J/√
2λπ and J is the spin) and evaluating the sum at each order in J using ζ- function regularization, we nd that all three cases actually give the same result. This is very nontrivial considering that our new summation prescription looks very dierent than the old one.
Furthermore, we show that this result agrees with the predictions of the Bethe ansatz. Thus, while the old summation prescription only seems to work when applied to the world-sheet frequencies of solutions with trivial support inAdS4, our summation prescription works more generally. Fully understanding why the old summation prescription breaks down for solutions with nontrivial support inAdS4warrants further study.
It would be useful to conrm our results using methods more rigorous than ζ-function regular- ization. This can be done using the contour integral techniques developed in [105], which can also be used to compute 1/J2n+1 and exponentially suppressed terms in the large-J expansion of the one-loop corrections. It would also be interesting to evaluate the one-loop correction to the spinning string energy in a way that does not rely on summation prescriptions. The basic idea would be
to identify the one-loop correction with a normal ordering constant which can be then determined by demanding that the quantum generators of certain symmetries preserved by the classical solu- tion have the right algebra. Something along these lines was done for the type IIB superstring in plane-wave background in [106]. Ultimately, fully understanding how to compute one-loop correc- tions to type IIA string theory in AdS4×CP3 may lead to important tests of the AdS4/CF T3
correspondence.
Appendix A
Dirac Algebras
A.1 Three-dimensional Dirac Algebra
The three-dimensional Dirac algebra is dened by
{γµ, γv}= 2gµν.
We chose the following representation
γ0=iσ2, γ1=σ1, γ2=σ3, where theσ's are the Pauli matrices
σ1=
0 1 1 0
, σ2=
0 −i i 0
, σ3=
1 0 0 −1
. (A.1)
The following relations are useful
[γµ, γv] = 2µνργρ, γµγνγµ=−γν,
γµγµ= 3, (γ·D) (γ·D) =D2+1
2γµνFµν,Ψ1Ψ2= Ψ2Ψ1, Ψ1ΓIJΨ2=−Ψ2ΓIJΨ1,
Ψ1ΓIJ KLΨ2= Ψ2ΓIJ KLΨ1,
Ψ1γµ1...µmΨ2= (−1)mΨ2γµm...µ1Ψ1.